Integral conditions for Hardy-type operators involving suprema
- Autores: Martin Krepela
- Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 68, Fasc. 1, 2017, págs. 21-50
- Idioma: inglés
- DOI: 10.1007/s13348-016-0170-6
- Texto completo no disponible (Saber más ...)
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Resumen
We characterize the validity of the weighted inequality \begin{aligned} \left( \int _0^\infty \Big [ \sup _{s\in [t,\infty )} u(s) \int _s^\infty g(x)\,\mathrm {d}x\Big ]^q w(t)\,\mathrm {d}t\right) ^\frac{1}{q} \le C \left( \int _0^\infty g^p(t) v(t)\,\mathrm {d}t\right) ^\frac{1}{p} \end{aligned} for all nonnegative functions g on (0,\infty ), with exponents in the range 1\le p<\infty and 0
Moreover, we give an integral characterization of the inequality \begin{aligned} \left( \int _0^\infty \Big [ \sup _{s\in [t,\infty )} u(s) f(s) \Big ]^q w(t)\,\mathrm {d}t\right) ^\frac{1}{q} \le C \left( \int _0^\infty f^p(t) v(t)\,\mathrm {d}t\right) ^\frac{1}{p} \end{aligned} being satisfied for all nonnegative nonincreasing functions f on (0,\infty ) in the case 0
